It was unclear whether the president’s acknowledgment that Mr. Lieberman was a finalist was intended to stoke the “Apprentice”-style frenzy of speculation he has favored with other high-profile picks, only to opt for a lesser-known candidate._________________Hiromichi

There is a TV show called "The Apprentice" in which people competed to be Donald Trump's assistant. He would eliminate one at a time on different shows. (You can look this up in Google. I never actually saw the show, so that's what I had to do!) It's hard to know the reasons why he leaks the names of people he is thinking about.